Lafforgue and Voevodsky Receive Fields Medals the rules for awarding the medals, Fields specified that the medals “should be of a character as purely international and im- personal as possible.” During the 1960s, in light of the great expansion of mathe- matics research, the possible number of medals to be awarded was increased from two to four. Today the Fields Medal is rec- ognized as the world’s highest honor in mathematics. Previous recipients are: Lars V. Ahlfors and Jesse Douglas (1936); Laurent Schwartz and Atle Selberg (1950); Kunihiko Laurent Lafforgue Vladimir Voevodsky Kodaira and Jean-Pierre Serre (1954); Klaus F. Roth and René Thom (1958); Lars Hör- On August 20, 2002, two Fields Medals were mander and John W. Milnor (1962); Michael F. awarded at the opening ceremonies of the Inter- Atiyah, Paul J. Cohen, Alexandre Grothendieck, national Congress of Mathematicians (ICM) in Bei- and Stephen Smale (1966); Alan Baker, Heisuke Hi- jing, China. The medalists are LAURENT LAFFORGUE and ronaka, Sergei P. Novikov, and John G. Thompson VLADIMIR VOEVODSKY. (1970); Enrico Bombieri and David B. Mumford The Fields Medals are presented in conjunction (1974); Pierre R. Deligne, Charles L. Fefferman, with the ICM, which is held every four years at dif- Grigorii A. Margulis, and Daniel G. Quillen (1978); ferent locations around the world. Although there Alain Connes, William P. Thurston, and Shing-Tung is no formal age limit for recipients, the medals have Yau (1982); Simon K. Donaldson, Gerd Faltings, traditionally been presented to mathematicians and Michael H. Freedman (1986); Vladimir Drinfeld, not older than 40 years of age, as an encourage- Vaughan F. R. Jones, Shigefumi Mori, and Edward ment for future achievement. The medal is named Witten (1990); Jean Bourgain, Pierre-Louis Lions, after the Canadian mathematician John Charles Jean-Christoph Yoccoz, and Efim Zelmanov (1994); Fields (1863–1932), who organized the 1924 ICM Richard Borcherds, William Timothy Gowers, Maxim in Toronto. At a 1931 meeting of the Committee Kontsevich, and Curtis T. McMullen (1998). of the International Congress, chaired by Fields, it The medals are awarded by the International was decided that the $2,500 left over from the Mathematical Union, on the advice of a selection Toronto ICM “should be set apart for two medals committee. The selection committee for the 2002 to be awarded in connection with successive In- Fields Medalists consisted of: James Arthur, Spencer ternational Mathematical Congresses.” In outlining Bloch, Jean Bourgain, Helmut Hofer, Yasutaka Ihara, 1264 NOTICES OF THE AMS VOLUME 49, NUMBER 10 H. Blaine Lawson, Sergei Novikov, George Papani- mathematics from Harvard University (1992). He colaou, Yakov Sinai (chair), and Efim Zelmanov. held visiting positions at the Institute for Advanced Study, Harvard University, and the Max-Planck- Laurent Lafforgue Institut für Mathematik before joining the faculty Laurent Lafforgue was born on November 6, 1966, of Northwestern University in 1996. In 2002 he in Antony, France. He graduated from the École Nor- was named a permanent professor in the School of male Supérieure in Paris (1986). He became a chargé Mathematics at the Institute for Advanced Study de recherche of the Centre National de la Recherche in Princeton, New Jersey. Scientifique (CNRS) (1990) and worked on the Arith- Vladimir Voevodsky made one of the most out- metic and Algebraic Geometry team at the Univer- standing advances in algebraic geometry in the sité Paris-Sud, where he received his doctorate past few decades by developing new cohomology (1994). In fall 2000 he was promoted to directeur theories for algebraic varieties. This achievement de recherche of the CNRS in the mathematics de- has its roots in the work of 1966 Fields Medalist partment of the Université Paris-Sud. Shortly there- Alexandre Grothendieck. Grothendieck suggested after he became a permanent professor of mathe- that there should be objects, which he called “mo- matics at the Institut des Hautes Études tives”, that are at the root of the unity between num- Scientifiques in Bures-sur-Yvette, France. ber theory and geometry. Laurent Lafforgue has made an enormous ad- One of the most important of the generalized vance in the Langlands Program by proving the cohomology theories is topological K-theory, de- global Langlands correspondence for function veloped chiefly by another 1966 Fields Medalist, fields. The Langlands Program, formulated by Michael Atiyah. An important result in topological Robert Langlands in the 1960s, proposes a web of K-theory is the Atiyah-Hirzebruch spectral se- relationships connecting Galois representations quence, developed by Atiyah and Friedrich Hirze- and automorphic forms. The influence of the Lang- bruch, which relates singular cohomology and topo- lands Program has grown over the years, with each logical K-theory. new advance hailed as an important achievement. For about forty years mathematicians worked The roots of the Langlands program are found hard to develop good cohomology theories for al- in one of the deepest results in number theory, the gebraic varieties; the best understood of these was Law of Quadratic Reciprocity, which was first the algebraic version of K-theory. A major advance proved by Carl Friedrich Gauss in 1801. This law came when Voevodsky, building on a little-under- allows one to describe, for any positive integer d, stood idea proposed by Andrei Suslin, created a the primes p for which the congruence theory of “motivic cohomology”. In analogy with the x2 ≡ d mod p has a solution. Despite many proofs topological setting, there is a relationship between of this law (Gauss himself produced six different motivic cohomology and algebraic K-theory. In proofs), it remains one of the most mysterious addition, Voevodsky provided a framework for facts in number theory. The search for general- describing many new cohomology theories for algebraic varieties. One consequence of Voevodsky’s izations of the Law of Quadratic Reciprocity stim- work, and one of his most celebrated achievements, ulated a great deal of research in number theory is the solution of the Milnor Conjecture, which for in the nineteenth century. Landmark work by Emil three decades was the main outstanding problem Artin in the 1920s produced the most general rec- in algebraic K-theory. This result has striking con- iprocity law known up to that time. One of the sequences in several areas, including Galois coho- original motivations behind the Langlands Pro- mology, quadratic forms, and the cohomology of gram was to provide a complete understanding of complex algebraic varieties. reciprocity laws. —Allyn Jackson The global Langlands correspondence for GLn proved by Lafforgue provides a complete under- standing of reciprocity laws for function fields. Lafforgue established, for any given function field, a precise link between the representations of its Ga- lois groups and the automorphic forms associated with the field. He built on work of 1990 Fields Medalist Vladimir Drinfeld, who in the 1970s proved the global Langlands correspondence for GL2 . Vladimir Voevodsky Vladmir Voevodsky was born on June 4, 1966, in Russia. He received his B.S. in mathematics from Moscow State University (1989) and his Ph.D. in NOVEMBER 2002 NOTICES OF THE AMS 1265.